Molecular Geometry & Bonding
what the shape decides
Series

Peierls distortion — the series

16 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A chain of 100 is more stable alternating. The electronic energy gained by alternating the bonds of a half-filled chain, the elastic cost of doing it, and their sum, all per site and all against the alternation. The sum has its best value away from zero, so the evenly spaced chain is not the stable structure.

    A chain cannot stay even

    Diagonalise a half-filled chain, alternate its bonds slightly, and diagonalise again. The electrons gain more than the springs lose, and they do so for every spring constant whatever — because the gain is steeper than a parabola near the origin and a logarithm beats any constant.

    part 1 · solids
  2. The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

    The gap is not the band width

    Two numbers describe a band and they answer different questions. The width is set by how many neighbours an atom has; the gap is set by how unequal they are. A structure can have a wide band and no gap, a narrow band and a large one, and changing one leaves the other alone.

    part 2 · solids
  3. d⁹: what a tetragonal distortion is worth. The electronic energy of d⁹, the elastic cost of the distortion, and their sum, against the fractional elongation of the axial bonds. The best distortion is at 0.14 and it is worth 0.54 in units of eσ; d⁶ in the same field gains 0, which is nothing.

    Copper is never quite octahedral

    A d⁹ ion in an octahedral field has three electrons in a doubly degenerate pair, which cannot be shared evenly. The energy it gains by distorting is linear in the distortion and the elastic cost is quadratic, so no stiffness holds the symmetric structure — and the four configurations with an even occupation gain exactly nothing.

    part 3 · applied
  4. Four fillings, four periods. four fillings of a ring of 120, and for each of them what a distortion of every available period is worth. Every period pays the same elastic cost, so the bars compare what the electrons give back and nothing else. The winner is one over the filling in every case.

    The distortion the filling chooses

    A half-filled chain of equal bonds is unstable and alternates — long, short, long, short. That is the case everyone is shown, and it is one case. Fill the chain a third of the way instead and the alternation is worthless: what wins is a pattern that repeats every three bonds, and the period is one over the filling at every filling tried.

    part 4 · solids
  5. The total energy against the distortion, at several gaps. The elastic cost plus the second-order lowering, for five gaps between the ground state and the excited state it mixes with. The critical gap is 1.00: above it the symmetric structure is the minimum, below it the minimum has moved off zero, and nothing about the molecule is degenerate in either case.

    A distortion needs two states

    A degenerate electronic state cannot survive — that is the Jahn–Teller theorem, and it can be computed. A closed shell can fail to survive too, and the condition is a number: the symmetric structure holds only while the nearest excited state of the right symmetry lies above 2λ²/k, and one of ten symmetry species in an octahedron is the right one.

    part 5 · applied
  6. An effect that cannot distort a molecule, deciding how far it distorts. The energy along one distortion coordinate, three times. With only the first-order term the minimum is at 0.6; with only the second-order term there is no minimum away from zero at all, because the gap of 1.5 is above the critical 1 for a closed shell on its own. With both, the molecule distorts to 1.06 — well past the first-order answer — and gains 0.09 more than the two separate stabilisations add up to.

    Two distortions in one coordinate

    A second-order Jahn–Teller effect that cannot distort a molecule by itself — its gap is half again above the critical value — nearly doubles the distortion when a first-order effect is already acting. The molecule goes to 1.075 instead of 0.600 and gains twice the energy, and it does it while the gap the second-order term divides by is opening rather than closing.

    part 6 · shape
  7. The whole difference lives at the two ends. The energy difference between the two dimerisations of an open chain, held at the same distortion, multiplied by the number of sites. It settles on a constant — 1.09 in units of the hopping — so the difference per site falls as one over the length, with a fitted exponent of -1. An end is a bond that is not there, and it is worth the same amount whatever it is attached to.

    The distortion the ends decide

    A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.

    part 7 · applied
  8. The chain distorts hardest where it stops. The alternation of each bond along a relaxed chain of 64, at four elastic constants, with the bulk value of each drawn as a dashed line. Every chain alternates more at its end than in its middle — by 1.21 times at the stiffest and 3.15 at the softest — and the excess dies away over a handful of bonds. The uniform alternation usually assumed is the flat part of these curves.

    The chain distorts hardest where it stops

    Holding the alternation uniform is what made the end energy a clean constant, and it is the one assumption the end-energy calculation had to make. Letting every bond find its own value shows the distortion is largest at the end and decays inwards over a measurable length — one and a half bonds in a strongly dimerised chain, five in a weak one.

    part 8 · solids
  9. Two lengths off the same chains, and only one of them has an exponent. The length an end's influence reaches into a chain of 320, against the gap the bulk has opened, over ten elastic constants and a factor of twenty in the gap. Fitted over the first twelve bonds — as a short-window fit does — the exponent is -0.476. Taken from the local decay rate extrapolated to a bond infinitely far from the end, it is -1.029, and every neighbouring pair of points gives between -1.06 and -0.91. The argument says −1. The two lines are the same ten profiles read two ways.

    A decay that keeps slowing down

    A healing length fitted over the first twelve bonds of each relaxed chain goes as the gap to the power −0.60, against an argument that says −1. A fitted exponent can belong to its window, and this one does — but not because the window is too short for the chain. The profile is not an exponential at all, so there is no single length for an exponent to be of.

    part 9 · solids
  10. The exponent against how far the tail can be seen. The fitted power against the reduced reach — how many coherence lengths of the decay survive above the floor before the excess is numerical noise. The six cases with a reach past six give a power between 0.41 and 0.57 and are drawn solid; the rest are hollow and their fitted powers are off this scale in the negative direction. The reach is not a choice — it falls as the gap closes, because the excess the tail starts from falls with it.

    The exponent was the floor

    Fitting the local decay rate against the reciprocal distance reads a power off the slope. It runs from 0.41 to 0.66 across ten stiffnesses and appears to settle near two thirds. It is not settling. The tail is dropping below the arithmetic's own floor sooner at every step, so each case's power is taken over a shorter piece of the curve than the last.

    part 10 · solids
  11. The exponent against where the fit is allowed to start. For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty. The standard start is six, by a rule of thumb — three coherence lengths — and the question was whether that choice is doing any work. Below six the exponent rises steeply; from six outward it is nearly flat. The rule of thumb sits on a plateau.

    The rule of thumb was on the flat part

    Fitting the Peierls tail discards the first few bonds of every profile, on a rule of thumb — three coherence lengths. Does that unexamined choice hide a second exponent? It does not. From six bonds outward the fitted power moves by half a per cent to nine; below six it moves seven times as much, and starting at two would have halved the very trend the fit reports.

    part 11 · solids
  12. The far end stops mattering, abruptly. The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty. Each curve is flat past a stiffness-dependent point and exactly flat past it — because beyond a profile's own reach there are no more local rates to add, so a larger window is the same fit. The standard sixty bonds is inside the flat part for every case.

    The other window was a plateau too

    Sweeping where the fit begins finds a plateau. The far end is the other window and nobody had swept it: inside each profile's own reach the exponent moves by at most 5.3 per cent, and past that reach every larger window returns exactly the same fit — because there are no more points to add. What the reach is depends on the stiffness, and for half the series it is an arbitrary rule rather than the physics.

    part 12 · solids
  13. Half the series stops on its own; the other half stops where it is told. How far from the chain's end the local decay rate can still be read, against how much of the chain the reading is allowed to cover. The dashed line is the cap itself. A curve that flattens below it has ended on the noise floor and the window never mattered; a curve that tracks the cap is being cut, and has more to say. The soft chains do the first and the stiff chains do the second.

    The window that was not a plateau

    The near and far ends of this fit both sit on plateaus, and it is tempting to expect the same of every window. The third choice — that the local decay rate is read from the first quarter of the chain — is not a plateau. It never bound the soft half of the series and it was setting the answer for the stiff half, where opening it moves an exponent by a seventh, always downward. And a profile allowed to end on its own always runs 13.26 of its own decay lengths, which is the ruler that shows three cases are still cut at the midpoint — four, once a stiff profile is counted only to the bond where its local rate turns back.

    part 13 · solids
  14. On an opened profile the local rate stops falling, and turns. The local decay rate at every bond, read to half the chain, for three stiffnesses. The soft chain's rate falls all the way to the last bond that produces one, where its excess meets the noise floor. The two stiff chains' rates reach a minimum and climb, because the excess is measured against a bulk value read from the chain's middle and the middle still carries both ends' healing. Every bond past the minimum is reading that reference, not the tail.

    The rate that turned back

    Opening the Peierls fit's window to half the chain added forty to eighty bonds to every stiff profile, and those bonds were read as tail. Most of them are not. The local decay rate on an opened stiff profile falls to a minimum and climbs again, because the bulk it is measured against is read from the chain's middle, which still carries both ends' healing — and a model of two ends and that reference, built from the quarter window alone, puts the minimum within two bonds of where it is. Read only to the turn, the fall of a seventh in the stiff exponents disappears, and one of the two cases the wider window was credited with resolving is not resolved.

    part 14 · solids
  15. Three bulks, and only one is the chain's. For each of the ten stiffnesses, how far three candidate bulk values sit from the bulk at the chain's own shift, on a logarithmic axis. A free infinite chain and a free ring of 320 sites are off by 10⁻⁵ to 10⁻³ at every stiffness — up to thirty thousand times the relaxation's floor of 10⁻⁸ — because neither has the shift the chain's fixed length imposes. The chain's own middle agrees with the shifted bulk to 10⁻¹¹ at every stiffness to 2.0 and then rises above it, by the two ends' tails, to 10⁻⁴ at the stiffest.

    A ring is not the bulk

    The local decay rate on a stiff Peierls chain turned back because its bulk was read from the chain's own middle. The remedy proposed was a ring of the same length, with no ends to heal from. A ring gets the soft chains wrong by thirty thousand times the floor — because the chain's relaxation holds its length fixed, and the ends' extra alternation is paid for by a shift in every bond that goes as one over the length. With that shift in the bulk, the soft middles agree to 10⁻¹¹, no stiff profile turns back, two tails describe every profile to its middle, and the stiff exponents keep rising where the published ones had turned down.

    part 15 · solids
  16. Freeing the chain's length leaves every tail exponent where it was. The tail exponent p of a two-ended fit A·b^(−p)·e^(−b/λ) from bond 6, at ten stiffnesses, for the chain whose total length is held fixed and for the chain at zero tension. The fixed chain's middle is compressed by an amount that goes as one over its length; the free chain's is not compressed at all. The exponents differ by at most 0.0009 — the constraint was never in them.

    The chain that pays for nothing

    Held to a fixed length, a Peierls chain pays for its ends' extra alternation with a compression of every bond. Let the length go and nothing is paid: the middle is the free infinite chain to twelve figures, the ends adjust locally — and every tail exponent comes out where the fixed chain put it, to under a thousandth. The constraint was never in the exponents. Nor is the drift in them a missing 1/b term: fitting one makes the exponent wander three times further.

    part 16 · solids

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